PLEASE SHOW ALL WORK. STEP BY STEP ON HOW TO SOLVE. THANKS Many caterpillars con

Question

PLEASE SHOW ALL WORK. STEP BY STEP ON HOW TO SOLVE. THANKS

Many caterpillars construct cocoons from silk, one of the strongest naturally occurring materials known. Each thread is typically 2.0 Hm in diameter and the silk has a Young's modulus of 4.0 x 109 N/m2 (a) How many strands would be needed to make a rope 10.75 m long that would stretch on 1.00 cm when supporting a pair of 80-kg mountain climbers? x 106 strands (b) Assuming that there is no appreciable space between the parallel strands, what would be the diameter of the rope? Cm Does the diameter seem reasonable for a rope that mountain climbers might carry? (Assume the diameter is reasonable if it is less than 5 cm.) Yes No Supporting Materials L PCast

Explanation / Answer

diameter of thread, d= 2.0 *10-6 m

Let total of "n" strands are need for the rope

Area of thread, a= (3.14)(d/2)2= (3.14)(2.0* 10-6/2)2 = (3.14* 10-12)

total Area of rope, A= na= n(3.14*10-12).................(1)

Also, total length of rope, L= 10.75 m

stretch in rope, l'= 1cm= 0.01m

Young's Modulus of silk, Y= 4 * 109 Nm-2

Force on rope, F= mg= (80)(9.8)= 784 N

ALso we know that Y= Stress/Strain= (F/A)/(l'/L)= FL/Al' ( where Stress= Force/ Area , and Strain= stretch/ original length)

Using given values and equation 1,

4*109= (784)(10.75)/(n* 3.14*10-12)(0.01)

n= 8428/(3.14 *10-14)(4* 109)

n= 671.01 * 105

n= 67.10 *106 starnds( ANS)

b). Using equation 1, A= n(3.14 *10-12)

Let diameter of rope is D then

(3.14)(D/2)2= 67.10 * 106 *3.14* 10-12

D2= 268.4 *10-6

D= 16.38 * 10-3 m = 1.68* 10-2m = 1.68 cm

c). Since the above calculated diameter (D=1.68cm) is less than 5 cm.

Hence Yes it is reasonable for mountain climbers.

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